Email: Thanks to everyone who wrote in about the maps. I've
received so much email that I may not be able to reply to everyone, but I
much appreciate all your comments and suggestions. Many of the things
people have been asking about are answered in this
list of frequently asked questions.

Maps of the 2012 US presidential election results

Election results by state

Most of us are, by now, familiar with the maps the TV channels and web
sites use to show the results of presidential elections. Here is a typical
map of the results of the 2012 election:

Click on any of the maps for a larger picture

The states are colored red or
blue to indicate whether a majority of their voters voted for the
Republican candidate, Mitt Romney, or the Democratic candidate, Barack
Obama, respectively. Looking at this map it gives the impression that the
Republican won the election handily, since there is rather more red on the
map than there is blue. In fact, however, the reverse is true – it
was the Democrats who won the election. The explanation for this apparent
paradox, as pointed out by many people, is that the map fails to take
account of the population distribution. It fails to allow for the fact
that the population of the red states is on average significantly lower
than that of the blue ones. The blue may be small in area, but they
represent a large number of voters, which is what matters in an election.

We can correct for this by making use of a cartogram, a map in which
the sizes of states are rescaled according to their population. That is,
states are drawn with size proportional not to their acreage but to the
number of their inhabitants, states with more people appearing larger than
states with fewer, regardless of their actual area on the ground. On such
a map, for example, the state of Rhode Island, with its 1.1 million
inhabitants, would appear about twice the size of Wyoming, which has half a
million, even though Wyoming has 60 times the acreage of Rhode Island.

Here are the 2012 presidential election results on a population cartogram
of this type:

As you can see, the states have been stretched and squashed, some of them
substantially, to give them the appropriate sizes, though it's done in such
a way as to preserve the general appearance of the map, so far as that's
possible. On this map there is now clearly more blue than red.

The presidential election, however, is not actually decided on the basis of
the number of people who vote for each candidate but on the basis of the electoral
college. Under the US electoral system, each state in the union
contributes a certain number of electors to the electoral college, who vote
according to the majority in their state. (Exceptions are the states of
Maine and Nebraska, which use a different formula that allows them to split
their electoral votes between candidates.) The candidate receiving a
majority of the votes in the electoral college wins the election. The
electors are apportioned among the states roughly according to population,
as measured by the census, but with a small
but deliberate bias in favor of less populous states.

We can represent the effects of the electoral college by scaling the sizes
of states to be proportional to their number of electoral votes, which
gives a map that looks like this:

This cartogram looks similar to the one above it, but it's not identical.
Wyoming, for instance, has approximately doubled in size, precisely because
of the bias in favor of small states.

The areas of red and blue on the cartogram are now proportional to the
actual numbers of electoral votes won by each candidate. Thus this map
shows at a glance both which states went to which candidate and which
candidate won more electoral college votes – something that you
cannot tell easily from the normal election-night red and blue map.

Election results by county

But we can go further. We can do the same thing also with the county-level
election results and the images are even more striking. Here is a
map of US counties, again colored red and blue to indicate Republican and
Democratic majorities respectively:

Now the effects we saw at the state level are even more pronounced: the red
areas appear overwhelmingly in the majority, an appearance again at odds
with the actual results of the election. Again, we can make a more helpful
respresentation by using a cartogram. Here is what the cartogram looks
like for the county-level election returns:

However, this map is still somewhat misleading because we have colored
every county either red or blue, as if every voter voted the same way.
This is of course not realistic: all counties contain both Republican and
Democratic supporters and in using just the two colors on our map we lose
any information about the balance between them. There is no way to tell
whether a particular county went strongly for one candidate or the other or
whether it was relatively evenly split.

One way to improve the map and reveal more nuance in the vote is to use not
just two colors, red and blue, but to use red, blue, and shades of purple
in between to indicate percentages of votes. Here is what the normal map
looks like if you do this:

And here's what the cartogram looks like:

As this map makes clear, large portions of the country are quite evenly
divided, appearing in various shades of purple, although a number of
strongly Democratic (blue) areas are visible too, mostly in the larger
cities. There are also some strongly Republican areas, but most of them
have relatively small populations and hence appear quite small on this map.

A slight variation on the same idea is to use a nonlinear color scale like
this:

These maps use a color scale that ranges from red for 70% Republican or
more, to blue for 70% Democrat or more. This is sort of practical, since
there aren't many counties outside that range anyway, but to some extent it
also obscures the true balance of red and blue.

Notes:

Frequently asked questions (FAQs): A list of frequently asked
questions concerning these maps, along with answers, can be found here.

County results: The county-level data I used came from here. They are, as far
as I know, up-to-date as of November 8, 2012, but a small number of
precincts still had not reported by that date, so a few results are
missing.

Thanks: Thanks to Robert Vanderbei for help with obtaining the
county-level data. Professor Vanderbei also suggested the purple maps, and
has made a terrific series of maps of his own, which you can find here.